Homoclinic orbits for second order Hamiltonian systems with potential changing sign. (English) Zbl 0867.70012

We study the second order Hamiltonian system \(\ddot q=-U'(t,q)\), where \(q:\mathbb{R}\to \mathbb{R}^N\) and \(U'(t,q)\) denotes the gradient with respect to \(q\) of a smooth potential \(U:\mathbb{R}\times \mathbb{R}^N\to\mathbb{R}\), \(T\)-periodic in time, having an unstable equilibrium point \(\overline{x}\) for all \(t\in\mathbb{R}\). Without loss of generality we can take \(T=1\) and \(\overline{x}=0\). Thus, \(q(t)\equiv 0\) is a trivial solution. We look for homoclinic orbits to 0, namely non-zero solutions of the problem \[ \ddot q=-U'(t,q), \qquad q(t)\to 0\text{ as }t\to\pm\infty, \qquad \dot q(t)\to 0\text{ as }t\to\pm\infty. \tag{P} \] The potential \(U\) has the form \(U(t,x)=-{1\over 2}x\cdot L(t)x+V(t,x)\), where \(L\) and \(V\) satisfy some technical assumptions. We prove that the problem (P) admits infinitely many solutions.


70H05 Hamilton’s equations
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems